Linear Physical Chains with Sturm-Liouville Characteristic Polynomials

Abstract

This paper deals with the existence and construction of linear physical chains whose characteristic (secular) polynomials are essentially one of four classical types: Hermite, generalized Laguerre, generalized Bessel, or Jacobi. Each of the results is useful for determining the natural frequencies (normal frequencies or eigenvalues) of the systems involved. A chain of coupled harmonic oscillators or the corresponding electrical analog can be regarded as a prototype of the systems under consideration. Chains of both arbitrary finite order and infinite order are considered. Let N be a prespecified positive integer. Consider a finite sequence {Sn}n=1N of linear dissipationless spring-mass chains in which Sn consists of masses m0, m1, …, mn−1 and springs with spring constants k0, k1, …, kn, and in which Sn+1 is obtained from Sn by attaching mass mn and spring with spring constant kn+1. Sn is connected to a wall by the spring having spring constant kn, but the chain may be free at the other end. In this case S1 is to consist only of mass m0 and spring with spring constant k1. Three major results relating to such a sequence are obtained. First, given a positive integer N, a procedure is developed whereby an existing Nth-order spring-mass system can readily be tested to determine whether the characteristic polynomials φ1, φ2, …, φN associated with chains S1, S2, …, SN are all classical polynomials of a single type; and if so, which type. The testing procedure can also be extended to the infinite-order case. Secondly, by means of large classes of examples, it is demonstrated that physical systems of any preassigned order N can actually be constructed so that for 1 ≤ n ≤ N the characteristic polynomials of Sn are all a specified one of the four classical types. Finally, it is shown that of the four possible kinds, only Jacobi- and Laguerre-type infinite-order systems can be generated; and the latter type can occur only if relatively stringent conditions on the physical parameters are satisfied.